Question

If \(g(x)=\frac{1}{x^{2}+1},\) find \(f(x)\) so that \(f(g(x))=\frac{x^{2}+1}{2}\)

Short answer

The function \( f(y) = \frac{1}{2y} \).

Step-by-step solution

Substitute and simplify

Substitute the expression for \(g(x)\) into \(f(g(x))\). Let \(y = g(x) = \frac{1}{x^2 + 1}\). The goal is to find \(f(y)\) to make \( f(g(x)) = f\left(\frac{1}{x^2 + 1}\right) = \frac{x^2 + 1}{2} \).

Replace the variable

Notice that, based on the substitution, \(y = \frac{1}{x^2 + 1}\). Therefore, \(x^2 + 1 = \frac{1}{y}\).

Express \(f(y)\) in terms of \(y\)

We want to express \(f(y)\) which equals \(\frac{x^2 + 1}{2}\), so substitute \(\frac{1}{y}\) for \(x^2 + 1\) in the denominator. Thus, \(\frac{x^2 + 1}{2} = \frac{\frac{1}{y}}{2} = \frac{1}{2y}\). Therefore, \(f(y) = \frac{1}{2y}\).

Key concepts

composition of functions

The composition of functions is the process of applying one function to the results of another. In this example, we have two functions, \(f(x)\) and \(g(x)\). We are given that \(g(x) = \frac{1}{x^2 + 1}\), and need to find \(f(x)\) so that \(f(g(x)) = \frac{x^2 + 1}{2}\). To solve this, we compose \(f\) and \(g\) as \(f(g(x))\). Considering \(g(x)\) is inside \(f\), we treat \(g(x)\) as a single entity and replace it where needed in \(f(x)\).
This is a common procedure in algebra, and helps to simplify and solve problems involving multiple functions. When we perform this operation, we basically chain the functions together.

substitution

Substitution is a method to simplify complex expressions by replacing a part of the expression with a single variable. Here, we started with the expression \( f(g(x)) = \frac{x^2 + 1}{2} \).
We know \( g(x) = \frac{1}{x^2 + 1} \) and we substitute \( y \) for \( \frac{1}{x^2 + 1} \), where \( y = g(x) \).
This makes it easier to handle the problem as we now deal with \(f(y)\) rather than the more complicated \(f(\frac{1}{x^2 + 1})\). The goal then becomes finding \(f(y)\) in terms of \(y\).
By using substitution, we simplify the expression and eventually find \(f(y) = \frac{1}{2y}\). This makes the process manageable and straightforward.

function simplification

Function simplification involves reducing a function to its simplest form. After substituting \(y\) for \(\frac{1}{x^2 + 1}\), we see that \(y = \frac{1}{x^2 + 1}\), which implies \(x^2 + 1 = \frac{1}{y}\).
Now, to simplify, we rewrite \(\frac{x^2 + 1}{2}\) using our substitution. It becomes \(\frac{\frac{1}{y}}{2}\), which further simplifies to \(\frac{1}{2y}\).
This step demonstrates the power of simplification. By breaking down complex expressions into simpler parts, solving them becomes much more manageable.
Simplification is a crucial step in problem-solving that reduces the cognitive load and reveals the solutions more clearly.

inverse functions

Inverse functions reverse the operation of a given function. If we have \(y = f(x)\), then the inverse function \(f^{-1}(y)\) gives us \(x\) in terms of \(y\).
In our example, we didn't explicitly find inverse functions, but the concept is related. We started with \(g(x)\) and found \(g^{-1}(y)\) implicitly.
We knew \(g(x) = \frac{1}{x^2 + 1}\), which we substituted as \(y = \frac{1}{x^2 + 1}\), leading us to \(x^2 + 1 = \frac{1}{y}\).
This substitution effectively utilized the idea of reversing \(g(x)\) to identify the relationship between \(x\) and \(y\).
Understanding inverse functions is essential when dealing with composition, as it allows us to 'undo' a function's operation, providing deeper insight into the problem's structure.